A379316 Positive integers whose prime indices include a unique squarefree number.

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211

Offset

1,1

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..58.

Example

The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Length[Select[prix[#], SquareFreeQ]]==1&]

Crossrefs

For all squarefree parts we have A302478, zeros of A379310.

Positions of 1 in A379306.

For no squarefree parts we have A379307, counted by A114374, strict A256012.

Partitions of this type are counted by A379308, strict A379309.

A000040 lists the primes, differences A001223.

A005117 lists the squarefree numbers, differences A076259.

A008966 is the characteristic function for the squarefree numbers.

A013929 lists the nonsquarefree numbers, differences A078147.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

Other counts of prime indices:

- A257991 odd, see A000041, A000070, A066207, A349158.

- A257992 even, see A000009, A038348, A066208, A379317.

- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.

- A330944 nonprime, see A000586, A000607, A076610, A330945.

- A379300 composite, see A023895, A034891, A036497, A302540, A379301.

- A379311 prime or 1, see A204389, A320629, A379312-A379315.

Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

Sequence in context: A103027 A093902 A269004 * A100475 A187921 A275059

Adjacent sequences: A379313 A379314 A379315 * A379317 A379318 A379319

Keyword

nonn

Author

Gus Wiseman, Dec 29 2024

Status

approved